Honestly I don't think that's much worse from a "reals are weird" point of view, although it is another wrinkle that there is a strict subset which is even harder to pin down.

Even worse, almost all real numbers aren’t definable

You can't define definability, because otherwise you could talk about the "least natural number not definable in fewer than n symbols" which is a contradiction for sufficiently large n.

Doesn’t definability just depend on the system you’re using to generate numbers being able to represent it?

For any system we could use to define a real number, (and not just because we are mortal), the system could only define or uniquely specify countably many real numbers. Any countable set of real numbers has measure 0, and therefore, almost all real numbers cannot be defined/specified using that system.

This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers

Or equivalently, from the natural numbers to the real numbers.

Hence dartboards have the mathematical property that a dart thrown into one is infinitely more likely to land on an integer than on a computable real (or a natural number, but for different reasons).

How so? Aren’t all integers computable real numbers?

Not in darts. You can prove this to your satisfaction, by attempting to reach the continuum between the integers.

The dart will bounce off the metal framey bit and stick into the forehead of local barfly, 'Very Angry Ron'. Then when you graph the result, violent discontinuities will appear in the function, across a wide range of scales.

These already have their own special set in category theory and are sometimes referred to, within the field, as; 'Assorted and Unreasonable Associates of Very Angry Ron'. That is, the other side of a reasonably big field, ideally with some cattle in the way.

Google returns this exact post to the query of “Assorted and Unreasonable Associates of Very Angry Ron”.

Must be another one of those pesky self-citing researchers.

Please forgive me father, for indulging in the sins of Onan.

That's ok, those ones aren't real.